powerShapeNA {shapeNA}

M-estimators of the Shape from the Power Family when Data is Missing

Description

Power M-estimators of shape and location were recently suggested in Frahm et al. (2020). They have a tuning parameter alpha taking values in [0,1]. The extreme case alpha = 1 corresponds to Tyler's shape matrix and alpha = 0 to the classical covariance matrix. These special cases have their own, more efficient functions tylerShapeNA and classicShapeNA, respectively. If the true location is known, it should be supplied as center, otherwise it is estimated simultaneously with the shape.

Usage

powerShapeNA(x, alpha, center = NULL, normalization = c("det", "trace", "one"),
         maxiter = 1e4, eps = 1e-6)

tylerShapeNA(x, center = NULL, normalization = c("det", "trace", "one"),
          maxiter = 1e4, eps = 1e-6)

classicShapeNA(x, center = NULL, normalization = c("det", "trace", "one"),
         maxiter = 1e4, eps = 1e-6)

Arguments

x

A data matrix or data.frame with missing data and p > 2 columns.

alpha

Tail index, a numeric value in the interval [0, 1]. Determines the power function. For more information see 'Details'.

center

An optional vector of the data's center, if NULL the center will be estimated simultaneously with the shape.

normalization

A string determining how the shape matrix is standardized. The possible values are

• 'det'such that the returned shape estimate has determinant 1.

• 'trace'such that the returned shape estimate has trace ncol(x).

• 'one'such that the returned shape estimate's top left entry (S[1, 1]) is 1.

maxiter

A positive integer, restricting the maximum number of iterations.

eps

A numeric, specifying tolerance level of when the iteration stops.

Details

These functions assume that the data were generated from an elliptical distribution, for Tyler's estimate this can be relaxed to generalized elliptical distributions The missingness mechanism should be MCAR or, under stricter distributional assumptions, MAR. See the references for details.

For multivariate normally distributed data, classicShapeNA is the maximum likelihood estimator of the location and scale. It is a special case of the power M-estimator with tail index alpha = 0, which returns the empirical covariance matrix and the empirical mean vector.

The function tylerShapeNA maximizes the likelihood function after projecting the observed data of each individual onto the unit hypersphere, in which case we obtain an angular central Gaussian distribution. It is a special case of the power M-estimator with tail index alpha = 1, which returns Tyler's M-estimator of scatter and an affine equivariant multivariate median according to Hettmansperger and Randles (2002).

The function powerShapeNA requires an additional parameter, the so-called tail index alpha. For heavy tailed data, the index should be chosen closer to 1, whereas for light tailed data the index should be chosen closer to 0.

Value

A list with class 'shapeNA' containing the following components:

S

The estimated shape matrix.

scale

The scale with which the shape matrix may be scaled to obtain a scatter estimate. If alpha = 1, then this value will be NA, as Tyler's shape matrix has no natural scale.

mu

The location parameter, either provided by the user or estimated.

alpha

The tail index with which the Power M-estimator has been called.

naBlocks

An naBlocks object, with information about the missingness of the data.

iterations

Number of computed iterations before convergence.

call

The matched call.

References

Frahm, G., & Jaekel, U. (2010). A generalization of Tyler's M-estimators to the case of incomplete data. Computational Statistics & Data Analysis, 54, 374-393. doi: 10.1016/j.csda.2009.08.019.

Frahm, G., Nordhausen, K., & Oja, H. (2020). M-estimation with incomplete and dependent multivariate data. Journal of Multivariate Analysis, 176, 104569. doi: 10.1016/j.jmva.2019.104569.

Hettmansperger, T. P., & Randles, R. H. (2002). A practical affine equivariant multivariate median. Biometrika, 89(4), 851-860. doi: 10.1093/biomet/89.4.851

See Also

powerShape, tylerShape and classicShape for the corresponding functions for data without missing values.

Examples

    ## Generate a data set with missing values
    x <- mvtnorm::rmvt(100, toeplitz(seq(1, 0.1, length.out = 3)), df = 5)
    y <- mice::ampute(x, mech = 'MCAR')$amp

    ## Compute some M-estimators
    res0 <- classicShapeNA(y, center = c(0, 0, 0))
    res1 <- powerShapeNA(y, alpha = 0.67, normalization = 'one')
    res2 <- tylerShapeNA(y, normalization = 'trace')

    ## Get location estimates
    res1$mu
    res2$mu
    ## Get shape estimates
    res0$S
    res1$S
    res2$S

    ## Print summary
    summary(res0)
    ## Inspect missingness pattern
    plot(res0$naBlocks)
    barplot(res0)